Explicit Traveling Wave Solutions to Nonlinear Evolution Equations

First of all, some technical tools are developed. Then the author studies explicit traveling wave solutions to nonlinear dispersive wave equations, nonlinear dissipative dispersive wave equations, nonlinear convection equations, nonlinear reaction diffusion equations and nonlinear hyperbolic equations, respectively.     

一些非线性发展方程的显式行波解

该文给出了一种构造非线性发展方程显式行波解的方法并用该方法得到了Hirota-Satsuma方程组,一类非线性常微分方程以及广义耦合标量场方程组的显式行波解.     

Generalized Extended Tanh-Function Method for Traveling Wave Solutions of Nonlinear Physical Equations

In this paper, the generalized extended tanh-function method is used for constructing the traveling wave solutions of nonlinear evolution equations. We choose Fisher's equation, the nonlinear schrödinger equation to illustrate the validity and advantages of the method. Many new and more general traveling wave solutions are obtained. Furthermore, this method can also be applied to other nonlinear equations in physics.     

非线性演化方程的孤立波解

用齐次平衡原则和辅助微分方程方法得到了6个重要的n次非线性演化方程的孤立波解.辅助微分方程方法的主要思想是借助简单的可解微分方程的解去构造复杂的非线性演化方程的行进波解.这里简单的可解微分方程称为辅助微分方程.本文使用的辅助方程有双曲正割幂型解或双曲正切幂型解.     

A Generalized （G＇/G）-Expansion Method to Find the Traveling Wave Solutions of Nonlinear Evolution Equations

In this article, we construct the exact traveling wave solutions for nonlinear evolution equations in the mathematical physics via the modified Kawahara equation, the nonlinear coupled KdV equations and the classical Boussinesq equations, by using a generalized （G＇/G）-expansion method, where G satisfies the Jacobi elliptic equation. Many exact solutions in terms of Jacobi elliptic functions are obtained.     

一类高阶非线性波方程的子方程与精确行波解

结合子方程和动力系统分析的方法研究了一类五阶非线性波方程的精确行波解.得到了这类方程所蕴含的子方程, 并利用子方程在不同参数条件下的精确解, 给出了研究这类高阶非线性波方程行波解的方法, 并以Sawada Kotera方程为例, 给出了该方程的两组精确谷状孤波解和两组光滑周期波解.该研究方法适用于形如对应行波系统可以约化为只含有偶数阶导数、一阶导数平方和未知函数的多项式形式的高阶非线性波方程行波解的研究.     

构造非线性微分差分方程精确解的一种方法

把最近提出的G′/G展开法推广到了非线性微分差分方程,利用该方法成功构造了一种修正的Volterra链和Toda链的双曲函数、三角函数以及有理函数三类涉及任意参数的行波解,当这些参数取特殊值时,可得这两个方程的扭状孤立波解、奇异行波解以及三角函数状的周期波解等.研究结果表明,该算法探讨非线性微分差分方程精确解十分有效、简洁.     

Similarity Solutions of Some Two-Space-Dimensional Nonlinear Wave Evolution Equations

Similarity reductions of the two-space-dimensional versions of the Korteweg-deVries, modified Korteweg-deVries, Benjamin-Davis-Ono, and nonlinear Schrödinger equations are presented, and some solutions of the reduced equations are discussed. Exact dispersive solutions of the two-dimensional Korteweg-deVries equation are obtained, and the similarity solution of this equation is shown to be reducible to the second Painlevé transcendent.     

Generalized Extended tanh-function Metho d for Traveling Wave Solutions of Nonlinear Physical Equations

In this paper,the generalized extended tanh-function method is used for constructing the traveling wave solutions of nonlinear evolution equations.We choose Fisher's equation,the nonlinear schr(o|¨)dinger equation to illustrate the validity and advantages of the method.Many new and more general traveling wave solutions are obtained.Furthermore,this method can also be applied to other nonlinear equations in physics.     

对称正则长波方程的行波解分岔

通过平面动力系统的方法讨论了对称正则长波方程的分岔问题.得到了该方程的分岔条件,在一些参数的具体值的情况下给出相图并通过微分方程的数值模拟方法模拟出了该方程的周期行波解、孤立行波解及无界行波解.     

Burgers-KdV方程的二类行波解

在熊树林关于“Burgers-KdV方程的一类解析解”工作的基础上,给出了Burgers-KdV方程的另外两类精确行波解,扩充了范恩贵等人工作的结果.     

强非线性波动方程孤子行波解

研究了一个强非线性波动方程.利用泛函分析变分迭代方法，首先构造了一个变分, 求出相应的Lagrange乘子；其次构造一个解的变分迭代, 选取初始孤子波；最后利用迭代方法依次求出各次孤子波的近似解.该方法是一个简单可行的近似求解非线性方程的方法     

非线性演化方程显式精确解的新算法

本文给出了一种求解非线性演化方程的新算法 .将这种算法运用于变形浅水波方程 ,获得了八组显式精确解 ,其中包括新的孤波解和周期解 .借助于 Mathematica软件 ,这种算法能够在 Computer上实现 .     

关于构造数学物理中非线性发展方程组孤波解的一种新算法

根据改进的sine-cosine法和吴文俊消元法,给出了一种构造非线性发展方程组孤波解的新算法。这种算法比已知的双曲函数法有更好的结论,并且在使用的过程中更简单。借助于MATH-EMATICA软件,这一算法能够在计算机上实现。     

变系数李方程组的精确行波解

利用修正的简单方程法对变系数李方程组进行求解,给出了变系数李方程组的双曲函数形式的行波解,当参数取特殊值时,便可以得到该方程组的精确孤波解.     

Traveling Wave Solutions to the Three-Dimensional Nonlinear Viscoelastic System

In this paper, we establish the existence of traveling wave solutions to the nonlinear three-dimensional viscoelastic system exhibiting long range memory. Under certain hypotheses, if the speed of propagation is between the speeds determined by the equilibrium and instantaneous elastic tensors, then the system has nontrivial trav- eling wave solutions. Moreover, the system has only trivial traveling wave solution in some cases.     

非线性波方程新的显式精确解

利用Darboux和一个可化为标准Bernoulli方程的4阶常微分方程,统一地处理了三个著名方程KdV方程,Kadomtsev-Petviashvili(KP)方程和Hirota-Satsuma(HS)方程的求解问题.给出了这些方程一批新的具有更为丰富形式的精确解,其中包括孤波解和行波解.     

Traveling Wave Solutions to the Free Boundary Incompressible Navier-Stokes Equations

In this paper we study a finite-depth layer of viscous incompressible fluid in dimension $n\ge 2$, modeled by the Navier-Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity-capillary effects, we allow for a second force field in the bulk and an external stress tensor on the free interface, both of which are posited to be in traveling wave form, i.e., time-independent when viewed in a coordinate system moving at a constant velocity parallel to the rigid lower boundary. We prove that, with surface tension in dimension $n\ge 2$ and without surface tension in dimension $n=2$, for every nontrivial traveling velocity there exists a nonempty open set of force and stress data that give rise to traveling wave solutions. While the existence of inviscid traveling waves is well-known, to the best of our knowledge this is the first construction of viscous traveling wave solutions. Our proof involves a number of novel analytic ingredients, including: the study of an overdetermined Stokes problem and its underdetermined adjoint problem, a delicate asymptotic development of the symbol for a normal-stress to normal-Dirichlet map defined via the Stokes operator, a new scale of specialized anisotropic Sobolev spaces, and the study of a pseudodifferential operator that synthesizes the various operators acting on the free surface functions. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

基于首次积分法对修正BBM方程的精确解的研究

基于首次积分理论,分析了两类修正的BBM方程的行波解,得到了它们的一些新解.